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GEOMETRICAL CORRELATION AND MATCHING OF 2D IMAGE SHAPES
Yu. V. Vizilter*, S. Yu. Zheltov
State Research Institute of Aviation Systems (GosNIIAS), 125319, 7, Viktorenko str., Moscow, Russia –
(viz, zhl)@gosniias.ru
Commission III, WG III/3
KEYWORDS: Image Matching, Image Comparison, Geometric Correlation, Mathematics
ABSTRACT:
The problem of image correspondence measure selection for image comparison and matching is addressed. Many practical applications
require image matching “just by shape” with no dependence on the concrete intensity or color values. Most popular technique for
image shape comparison utilizes the mutual information measure based on probabilistic reasoning and information theory background.
Another approach was proposed by Pytiev (so called “Pytiev morphology”) based on geometrical and algebraic reasoning. In this
framework images are considered as piecewise-constant 2D functions, tessellation of image frame by the set of non-intersected
connected regions determines the “shape” of image and the projection of image onto the shape of other image is determined.
Morphological image comparison is performed using the normalized morphological correlation coefficients. These coefficients
estimate the closeness of one image to the shape of other image. Such image analysis technique can be characterized as an “intensity-
to-geometry” matching. This paper generalizes the Pytiev morphological approach for obtaining the pure “geometry-to-geometry”
matching techniques. The generalized intensity-geometrical correlation coefficient is proposed including the linear correlation
coefficient and the square of Pytiev correlation coefficient as its partial cases. The morphological shape correlation coefficient is
proposed based on the statistical averaging of images with the same shape. Centered morphological correlation coefficient is obtained
under the condition of intensity centering of averaged images. Two types of symmetric geometrical normalized correlation coefficients
are proposed for comparison of shape-tessellations. The technique for correlation and matching of shapes with ordered intensities is
proposed with correlation measures invariant to monotonous intensity transformations. The quality of proposed geometrical correlation
measures is experimentally estimated in the task of visual (TV) and infrared (IR) image matching. First experimental results
demonstrate competitive quality and better computational performance relative to state-of-art mutual information measure.
1. INTRODUCTION*
Image matching is one of actual problems in remote sensing and
machine vision. The selection of image matching procedure for
some practical application depends on different aspects of the
whole matching algorithm. Some of these aspects concern with
discontinuities in object space and perspective distortions of
compared images. Other aspects deal with sub-pixel precision
and computational cost of matching. Many researchers have
explored these aspects. We have investigated them too in our
previous works (Zheltov, Sibiryakov, 1997; Zheltov, Vizilter,
2004). This paper concerns the other aspect of matching
procedure – the measure of image fragments similarity (or
dissimilarity) to be utilized in matching algorithm.
In some practical cases the final quality and performance of
image matching procedure essentially depends on quality and
performance of image comparison function. So, there are some
special functions for comparison of binary, grayscale, colored,
multispectral and other types of images. This paper devoted to
the problem of image matching “just by shape” with no
dependence on the concrete intensity or radiometric pixel
values. For example, one can compare images of one scene
captured at different seasons, different time of day, in different
weather and lighting conditions, in different spectral ranges and
so on. The most popular technique for such image shape
comparison utilizes the mutual information measure based on
probabilistic reasoning and information theory background
(Maes, 1997). The other approach was proposed by Pytiev (so
called “Pytiev morphology”) based on geometrical and
algebraic reasoning (Pytiev, 1993). This paper develops the
ideas of this geometrical approach.
* Corresponding author.
In the framework of Pytiev morphology images are considered
as piecewise-constant 2D functions. The tessellation of image
frame by the set of non-intersected connected regions with
constant intensities determines the “shape-tessellation” or
simply “shape” of the image. Morphological image comparison
is performed using the normalized “morphological” correlation
coefficients.
The motivation of reported theoretical research is following. In
Pytiev approach one supposes that the “shape” F of some “class
sample” f is given by supervisor, and one needs to determine
whether any observed image g belongs to this class. This
scheme presumes that the shape of f is more believable than the
shape of g. But in practice it is not always reasonable. A lot of
applications require estimating the closeness or difference
between two images with the same belief of shape. So, one
needs to suppose that both f and g are randomly selected from
their “shape sets” F and G correspondingly. In such case they
should be compared either as images (“intensity comparison”
like in linear correlation) or as pure shapes (“geometrical
comparison”). The intermediate “intensity-geometrical”
comparison like Pytiev correlation is not reasonable here.
Besides, the geometrical shape comparison could provide the
possibility of image matching for images segmented not only by
intensity, but by other features, for example, by texture. Finally,
it is useful to design some symmetric image comparison
measures invariant to shape preserving intensity transforms.
According to this motivation some new correlation tools for
geometrical comparison and matching of shape-tessellations
will be proposed based on generalization of Pytiev
morphological correlation. Destination and properties of
proposed geometrical correlation tools will be comparable and
competitive with those of mutual information.
ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume I-3, 2012 XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
191
2. IMAGE COMPARISON PROBLEM
AND RELATED WORKS
Let image is represented by 2D intensity function f(x,y): R,
R2, where is a frame region of bounded size, R – set of
real numbers, R2 – 2D plane of image. The operation of
“comparison” of images f(x,y) and g(x,y) presumes the
estimation of their “similarity” or, conversely, “dissimilarity”.
Numerical estimation of “similarity” is usually performed by
such similarity measures as correlation coefficients.
“Dissimilarity ” is estimated by distances in some metrics. The
proper formulas of such coefficients and distances are
determined by image models and image spaces we use.
Classical approach to image comparison considers images as
elements of Hilbert space L2() with scalar product (f,g),
Euclidean norm || f ||2 = (f,f) and Euclidean distance
dE(f,g) = || f – g ||. But photographic linear transforms of image
intensity can essentially change the distance between images
and even the pair of very similar images will seem to be very
“far”. So, for image comparison invariant relative to linear
intensity transform the following normalized linear correlation
coefficient is applied
||||||||
),(),(
gf
gfgfKN .
In this formula images are usually centered, i.e. decreased by
their constant average values.
Such approach is well-known in photogrammetry as a least
squares matching (Gruen, Baltsavias, 1985).
But even the normalized linear correlation is not invariant
relative to more complex (nonlinear) intensity transforms
occurred in practice. For image comparison under such complex
and unknown intensity transforms the mutual information
approach was proposed (Maes, 1997). Mutual information
I(A,B) estimates the dependence of two random variables A and
B by measuring the distance between the joint distribution
pAB(a,b) and the distribution of complete independence
pA(a)pB(b) by means of following expressions
I(A,B) = H(A) + H(B) – H(A,B),
H(A) = – a pA(a) log pA(a),
H(B) = – b pB(b) log pB(b),
H(A,B) = – a b pAB(a,b) log pAB(a,b),
where H(A) is an entropy of A, H(B) is an entropy of B, and
H(A,B) is their joint entropy. For two image intensity values a
and b of a pair of corresponding pixels in the two images,
required empirical estimations for the joint and marginal
distributions can be obtained by normalization of the joint (2D)
and marginal (1D) histograms of compared image fragments.
Maximal I(A,B) value corresponds to the best geometrical
matching of image fragments.
This approach is based on probabilistic reasoning. It provides
the robust tool for matching of images with different intensities
based on their joint 2D histograms. But these histograms can
not explain the geometrical idea of image “shape” in some
evident form. Such mathematical “shape” formalism is given in
evident form in the so-called “morphological” approach for
image comparison proposed by Pytiev (Pytiev, 1993).
In the framework of this approach images are considered as
piecewise-constant 2D functions
f(x,y) = i=1,..,n fi Fi(x,y),
where n – number of non-intersected connected regions of
tessellation F of the frame , F={F1,…,Fn}; f=(f1,…,fn) –
corresponding vector of real-valued region intensities;
Fi(x,y){0,1} – characteristic (support) function of i-th region:
.,0
;),(,1),(
otherwise
Fyxifyx
i
Fi
Set of images with the same tessellation F is a convex and close
subspace FL2() called shape-tessellation or simply shape:
F = { f(x,y) = i=1,..,n fi Fi(x,y), fRn}.
For any image g(x,y)L2() the projection onto the shape F is
determined as
gF(x,y) = PF g(x,y) = i=1,..,n gFi i(x,y),
gFi = (Fi,g) / || Fi ||2, i=1,…,n.
Here PF is a projective operator called projector to F. Let’s note
that for any image f(x,y) the description of its “shape” by a set
of frame tessellation regions F={Fi}, by corresponding vector
subspace F and by corresponding projection operator (“shape
projector”) PF are practically equivalent.
Pytiev morphological comparison of images f(x,y) and g(x,y) is
performed using the normalized morphological correlation
coefficients of the following form
,||||
||||),(
g
gPFgK F
M .||||
||||),(
f
fPGfK G
M
The first formula estimates the closeness of image g to the
“shape” of image f. Second formula measures the closeness of
image f to the “shape” of image f. In general KM(g,F) is not
equal to KM(f,G). So, this morphological image matching score
is asymmetric in contrary to symmetric linear correlation
coefficient KN(f,g).
3. GENERALIZED INTENSITY-GEOMETRICAL
CORRELATION
While the linear correlation coefficient KN(f,g) compares images
just as intensity functions (“intensity-to-intensity” matching),
morphological coefficient KM(f,G) compares the intensity
distribution of f(x,y) with geometrical shape of g(x,y)
(“intensity-to-geometry” matching). In other words, there are
different types of image correlation: intensity and intensity-
geometrical correlation. Are these correlation techniques
principally independent or could be united in some more
generic correlation frame work? In this section the answer will
be given.
Let some images f and g are given and some shape W is
considered. Then one can measure the angle (fW gW). The value
ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume I-3, 2012 XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
192
(fW,gW) characterizes the similarity of images g and f relative to
shape W. In particular, if cos(fW gW)=0, then images f and g are
orthogonal or independent relative to shape W.
This allows introducing the normalized linear correlation
coefficient of images g and f relative to shape W:
||||||||
),(),,(
gf
gfWgfK WW
N .
Following expressions are satisfied:
);,(||||||||
),(
||||||||
),(),,( gfK
gf
gf
gf
gfGFgfK N
GFGFN
);,(||||
||||
||||||||
),(),,( 2
2
2
GfKf
f
ff
ffGffK M
GGGN
).,(||||
||||
||||||||
),(),,( 2
2
2
FgKg
g
gg
ggFggK M
FFFN
So, the generalized intensity-geometrical correlation coefficient
KN(f,g,W) includes both the linear correlation coefficient and the
square of Pytiev correlation coefficient as its partial cases. It
means that the intensity-geometrical (morphological)
correlation is more generic relative to intensity (linear)
correlation and includes it as a part of its own framework.
Note that in this generic form KN(f,g) “structurally corresponds”
to the square of morphological coefficient KM2(f,G). Hence, if
one needs to compare the sensitivity of different correlation
techniques, one should compare KN with KM2, but not with KM.
4. GEOMETRICAL SHAPE COMPARISON BASED ON
STATISTICAL AVERAGING OF PROJECTED IMAGES
If one fixes some shape G and takes some different images f
from shape F, there will be different values of KM2(f,G).
Nevertheless, one can try to average these values over a set of
images from F and obtain some “average effective”
morphological correlation coefficient for comparison of shapes
F and G. Let f(x,y) form F is a piecewise-constant 2D function
described above (in section 2). Image g(x,y) from G is an
analogous 2D function with m as a number of tessellation
regions G={G1,…,Gm}; g=(g1,…,gm) – vector of intensity
values; Gj(x,y){0,1} – support function of j-th region. Let’s
introduce following additional set of “S-variables”:
S – area of the whole frame ;
Si = || Fi(x,y) ||2 – area of tessellation region Fi;
Sj= || Gj(x,y) ||2 – area of tessellation region Gj;
Sij= ( Fi(x,y), Gj(x,y) ) – area of intersection FiGj.
In terms of mutual information approach these S-variables
correspond to following marginal and joint probabilities:
pF(fi) = Si /S, pG(gj) = Sj /S, pFG(fi,gj) = Sij /S.
With account of these S-variables
|| f ||2 = i=1,..,n fi2 Si,
|| fG ||2 = j=1,..,m fGj2 Sj,
where fGj = (i=1,..,n fi Sij)/Sj, j=1,..,m.
Additionally introduce the following assumptions about the
distributions of probability densities for intensity values f1,…,fn:
1) p(f1,…,fn) = p(f1)…p(fn) – values f1,…,fn are independent in
general;
2) p(f1) = … = p(fn) – values f1,…,fn are equally distributed;
3) i=1,..,n: p(fi) = p(–fi) – values f1,…,fn are distributed
symmetrically relative to 0.
Consequently, the expectation of all intensity values in this
model is zero-valued: fi =0, and the covariation has the form
,,0
;,2
otherwise
kiifff ki
where – is a dispersion of probability distribution p(fi).
With account of introduced assumptions the mean square of
norm for image f of shape F has a form
|| f ||2 = i=1,..,n fi2 Si = i=1,..,n
2 Si = 2 i=1,..,n Si = 2 S.
The mean square of projection norm for image f of shape F and
fixed shape G has a form
|| fG ||2 = j=1,..,m fGj2 Sj = j=1,..,m (i=1,..,n fi Sij)
2/Sj2 Sj =
= j=1,..,m (i=1,..,n 2 Sij
2) / Sj = 2 j=1,..,m i=1,..,n Sij2 / Sj .
Let’s determine the mean square effective morphological
correlation coefficient (MSEMCC) for shapes F and G as
.||||
||||),(
2
22
f
fGFK G
M
After evident substitutions MSEMCC takes the following form
mj nij
ijij
MS
S
S
SGFK
,...,1 ,...,1
2 ),(
= j=1,..,m i=1,..,n K(Fi,Gj) KM2(Gj,Fi), (1)
where K(Fi,Gj) = Sij / S – normalized influence coefficient for
pair of regions Fi and Gj;
KM2(Gj,Fi) = Sij / Sj – square of normalized morphological
correlation for pair of regions Fi and Gj of the form
KM(A,B) = || PB A || / || A || = || A B || / || A ||;
KM(F,G) [0,1] – normalization property;
KM(F,F) = 1 – identity property.
However there is one important problem with MSEMCC (1). It
can not take in account the “independence” of shapes in a
Pytiev sense. For such independent shapes F and G for any
centered images fF, gG: KM(g,F) = KM(f,G) = 0, but
MSEMCC can not provide the property KM(F,G) = KM(G,F) = 0
due to the fact that intensity of images in our reasoning above
was not supposed to be centered.
Let’s start from the centered Pytiev morphological coefficient
ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume I-3, 2012 XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
193
,||||
||||),(
0
00
ff
ffPGffK G
M
where f0 = i=1,..,n fi Si / S – average intensity value for image f.
Introduce the mean square centered morphological correlation
coefficient (MSCMCC) for shapes F and G as
.||||
||||),(
2
0
2
02
ff
ffGFK G
MC
After some substitutions MSCMCC takes the following form
.
1
),(
,...,1
2
,...,1 ,...,1
2
2
ni
i
mj ni
i
j
ijj
MC
S
S
S
S
S
S
S
S
GFK (2)
Expression (2) is not so compact as (1), but due to relation
ni
i
ni
iM
MC
S
S
S
SGFK
GFK
,...,1
2
,...,1
2
2
2
1
),(
),(
MSCMCC provides both properties of (1) and important
additional property
KMC(F,G) = 0, KMC(G,F) = 0
fF, gG: || PF g – g0 || = || PG f – f0 || = 0.
This property describes the independence of shapes in a Pytiev
sense (Pytiev, Chulichkov, 2010).
5. SYMMETRIC GEOMETRICAL SHAPE
CORRELATION COEFFICIENTS
Let’s try to construct symmetric modification of MSEMCC (1).
Symmetric pair correlation coefficient for regions Fi and Gj can
be defined as follows:
,),(2
2
ijji
ij
ji
ji
jiMSSSS
S
gf
gfGFK
Then modifying the expression (1) one can obtain the following
symmetric geometrical correlation coefficient (SGCC):
mj niijji
ij
MSSSS
S
SGFK
,...,1 ,...,1
21
),(
= j=1,..,m i=1,..,n K(Fi,Gj) KMS(Gj,Fi). (3)
SGCC satisfies both properties of (1) and symmetry property
KMS(F,G)=KMS(G,F).
The other symmetric modification of geometric correlation (1)
can be proposed based on the linear normalized pair correlation
coefficient for regions Fi and Gj:
,||||||||
),(),(
ji
ij
GF
GF
jiNSS
SGFK
ji
ji
Geometrical linear correlation coefficient (GLCC) can be
defined as
mj niji
ij
NSS
S
SGFK
,...,1 ,...,1
21
),(
= j=1,..,m i=1,..,n K(Fi,Gj) KN(Gj,Fi). (4)
Properties of GLCC are the same as the SGCC properties.
6. COMPARISON OF SHAPES WITH ORDERED
INTENSITIES
Finally let’s consider the problem of comparison of shapes with
ordered intensities. This case corresponds to the class of
monotonous intensity transforms. For simplicity we suppose
that there are no equal values in intensity vectors f and g.
For each region Fi one can define the relative intensity function
;),(,1
;),(,0
;),(,1
),(
i
i
i
F
fyxfif
fyxfif
fyxfif
yxi
and each region Gj has the analogous function Gj(x,y).
The similarity of ordered intensity sequences in “ordered
shapes” F+ and G+ related to pair of regions Fi and Gj can be
described by pair relative intensity correlation coefficient
.||||||||
),(),(
ji
ji
GF
GF
ji GFK
The use of this pair coefficient allows easy modifying the
proposed geometrical correlation coefficients (1), (3) and (4) in
order to take in account the ordering of intensity values.
For example, the modified geometrical linear correlation
coefficient (MGLCC) will have a form
KN+(F+,G+) = j=1,..,m i=1,..,n K(Fi,Gj) KN(Gj,Fi) K(Fi,Gj).
Following properties of MGLCC are satisfied:
ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume I-3, 2012 XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
194
KN+(F+,G+) [-1,1];
KN+(F+,F+) = 1;
KN+(-F+,F+) = -1;
KN+(F+,G+)=KN
+(G+,F+).
Here “-F+” means the shape with the same frame tessellation as
F+, but inverse order of region intensities. In analogous way the
correlation coefficients (1) and (3) can be modified too for
obtaining their intensity ordering sensitive modifications.
7. EXPERIMENTAL RESULTS
Proposed geometrical correlation techniques were implemented
in software and tested over a set of real images including remote
sensing and multispectral images. The quality of proposed
geometrical correlation measures was investigated relative to
the quality of mutual information measure and Pytiev
correlation coefficient in the task of TV (visual band) and IR
(infrared band) image matching. TV and IR images were
geometrically co-registered. Then some fragments of TV image
were compared with all equal-sized fragments of corresponding
IR image. For increasing of matching robustness the histograms
of these TV and IR fragments were segmented with n=m=4
levels using least square optimal segmentation procedure. For
segmented image fragments f (TV) and g (IR) at each fragment
g position (x,y) following similarity measures were calculated:
- mutual information MI = I(F,G);
- square of centered Pytiev morphological coefficient
Kp = KM2(f – f0,G);
- square of MSEMCC Km = KM2(F,G) (1);
- square of MSCMCC Kmc = KMC2(F,G) (2);
- SGCC Kms = KMS(F,G) (3);
- SLCC Kn = KN(F,G) (4).
The quality of these measures was estimated by following
statistics of 2D correlation function С(x,y) – SNR (signal-to-
noise ratio) and E (exceeding of first maximum to second one):
,||
||;
||
2
11
C
CE
CSNR
where C1 – global maximum of correlation value; C2 – second
global maximum of correlation value out of some small
neighborhood of first global maximum; μ – mean value of
correlation function; σ – dispersion of correlation function.
Figure 1 demonstrates the example (Example 1) of experimental
TV-IR data corrupted by Gaussian noise. Figure 2 demonstrates
2D-graphs (correlation fields) of different correlation measures
for matching of TV etalon and IR image (for Example 1).
Corresponding numerical data are listed in Table 1. Example 2
(Figures 3, 4; Table 2) presents the other TV-IR image pair
corrupted by Gaussian noise with smaller size of etalon
fragment.
Analysis of these and other analogous TV-IR matching
examples allows making following conclusions:
1. In case of relatively large fragment matching for clear and
noisy images both Pytiev morphological coefficient and all
proposed geometrical correlation measures (1)-(4) provide SNR
and E values close to mutual information characteristics (a little
bit worse or better).
2. In case of relatively small fragment matching both for clear
and noisy images proposed asymmetrical geometrical
correlation measures (1)-(2) provide SNR and E values slightly
better than Pytiev morphological coefficient and mutual
information characteristics. But proposed symmetrical
geometrical correlation measures (3)-(4) with small fragment
and essential noise provide SNR and E values essentially worse
than other tested measures and even can fail in global peak
position (like in Example 2). These measures are not robust
enough relative to noise and fragment size.
3. Both shapes of correlation fields and SNR and E values for
proposed measures (1) and (2) are exactly equal in all
experiments due to the linear relation pointed in section 4 (but
they will be not equal in case of inverse order of etalon and test
shape arguments).
Measure Max value SNR E
MI 0.00865*10-3 4.7334 2.7357
Kp 0.01159 4.7320 2.9902
Km 0.25491 4.7200 2.7725
Kmc 0.00384*10-3 4.7200 2.7725
Kms 0.14647 4.8785 2.5354
Kn 0.25548 4.8587 2.3340
Table 1. Numeric data for TV-IR matching (Example 1).
Measure Max value SNR E
MI 0.03613 9.2157 1.7512
Kp 0.04361 8.9576 1.6643
Km 0.28774 10.3510 1.8913
Kmc 0.00159*10-3 10.3510 1.8913
Kms 0.1599 4.6429 1.2303
Kn 0.27794 4.1009 1.2635
Table 2. Numeric data for TV-IR matching (Example 2).
Totally, we can resume first experimental results as follows.
Proposed MSEMCC and MSCMCC measures are very close to
mutual information by matching characteristics and shape of
correlation field. In our experiments these measures are usually
a little bit better than MI (not statistically proved). But, taking in
account the computational simplicity of expression (1) relative
to MI and MSCMCC formulas, one can conclude that
MSEMCC is the best geometrical correlation measure for
practical applications with matching of different band images.
a)
b)
c)
d)
Figure 1. Example 1: a) etalon TV fragment; b) test IR image;
c) segmented TV fragment; d) corresponding segmented IR
fragment.
ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume I-3, 2012 XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
195
Figure 2. Correlation fields for TV-IR matching (Example 1).
a)
b)
c)
d)
Figure 3. Example 2: a) etalon TV; b) test IR; c) segmented TV
fragment; d) corresponding segmented IR fragment.
CONCLUSION
This paper generalizes some results of Pytiev morphological
image analysis (Pytiev, Chulichkov, 2010).
Following theoretical results of this paper are new and original:
- The generalized intensity-geometrical correlation coefficient is
proposed that includes the linear correlation coefficient and the
square of Pytiev correlation coefficient as its partial cases.
- The mean square effective morphological correlation
coefficient (MSEMCC) is proposed based on the statistical
averaging of projected images with the same shape. Mean
square centered morphological correlation coefficient
(MSCMCC) is obtained under centering for projected images.
- Two types of symmetric geometrical normalized correlation
coefficients are proposed based on the form of MSEMCC.
- The technique for correlation and matching of shapes with
ordered intensities is proposed too. It is invariant just relative to
monotonous intensity transformations.
Figure 4. Correlation fields for TV-IR matching (Example 2).
First experimental testing of proposed similarity measures is
performed for TV and IR image matching task. Based on these
experiments, MSEMCC seems to be competitive in quality and
better in computational performance relative to the state-of-art
mutual information measure. But proposed symmetric
correlation measures are not stable enough relative to noise and
should be improved.
Further research will include the design of more robust
symmetric geometrical correlation coefficients and massive
experimental exploration of precision and robustness for all
described shape correlation techniques.
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Maximization of Mutual Information. IEEE Transactions on
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Pyt’ev Yu., 1993. Morphological Image Analysis. Pattern
Recognition and Image Analysis. V.3. No 1, pp. 19-28.
Pyt’ev Yu., Chulichkov A., 2010. Methods of Morphological
Image Analysis. Fizmatlit, Moscow, 336 p. (in Russian)
Zheltov S., Sibiryakov А. 1997. Adaptive Subpixel Cross-
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Zheltov S., Vizilter Yu., 2004. Development of effective
procedures for automatic stereo matching. // International
Archives of the Photogrammetry, Remote Sensing and Spatial
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723.
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